I am trying to be purposeful and organized in taking the "Calls to Action" from the recent NCTM conference in San Francisco. I am going to play with "Deleting the Texbook" over my next few blog posts (watch Meyer's most recent NCTM presentation here for elaboration). Since I attended my first conference about 7 years ago, I have been heavily influenced by the work of Dan Meyer. His Ted Talk "Math Class Needs a Makeover" really changed the way I thought about teaching math.

​I want to share something I did with first as a math coach with a class of grade 3 students and then as a presentation for teachers at the Nova Scotia MTA conference a few year back. I feel like I have been taking this particular "Call to Action" for many years now; I am excited about the evolution of this thinking and I wanted to show how I started with this work.

The students in this class were a few weeks out from writing a provincial math assessment. The classroom teacher wanted to do some questions from the practice booklet. This was the original question:

Instead of posing the question as is, I took out as much information as possible while keeping the math I wanted the students to learn in mind. The students needed to get curious before I hit them with the math.

All staff at schools need to wear an ID badge and laynard (wait for it...it will make sense in a moment). Lots of people keep keys, change, notes, etc in the plastic badge holder. Instead of asking the above question, I announced to the class that I had some change in my ID holder (I put the correct amount in and shook it for dramatic affect!). Their job was to figure out how much money I had. They were a little confused at first, so I prompted them; I told them they could ask me questions about the coins. I gave them a few minutes to think, then to confer with a partner before I took questions from the class. I recorded and answered their questions. Here are the questions they came up with along with my answers in brackets:

How much money do you have? (I wouldn't answer that one!)

How many coins to you have? (8)

What types of coins do you have? (quarters, nickels and dimes)

How many of each? (half are quarters, one quarter are dimes and the rest are nickels)

Each time I took a question, I asked students "Why do you want to know that?" or "How will that information help you solve the problem?"

Once they had all the information they felt they needed, they set off to solve the problem. They basically constructed the original question on their own. And because they had a part in building it, they really understood the context of the problem and had ideas of how they might solve it. When they were confident that they had the solution, I let them empty my ID badge and count the coins. That is much more satisfying than just giving the correct answer. Students got to prove that they were right, rather than the teacher being in charge of all the answers.

I often hear from teachers that their students don't understand word problems and that it's not the math, it's reading comprehension. I see this strategy of building a question with the class as a way to help them get inside the problem before they start. Another reason I like this strategy so much is that I was able to create it by modifying a resource (practice booklet) that I already had. We can get bogged down trying to find appropriate resources for students at appropriate levels. By using a resource I already had, taking out parts, and layering them back in as needed I was able to differentiate on the fly. No tiered lesson plan, no separate handouts. It was one problem that we worked together as a class to first build, then solve.

The Most Interesting Question I Almost Overlooked

"What kind of coffee do you drink?

I didn't understand where the student was going with the question but I asked the usual questions of him: "How will that help you solve the problem?" and "Why do you want to know?"

The student thought I probably kept my coffee money in my ID badge. If he knew the coffee I drank, he would have the problem solved.** Based on his question, we created a series of extension questions to keep the problem solving going.

I get so excited when my kids tell me stories of what is happening in their math classes. This is a favourite.

My youngest son (age 7, grade 2) began his story as soon as I picked him up from school.

"Mommy, did you know that if you wanted to buy, let's say some....fabric, you couldn't just like go the fabric store and say 'I'll have 20 pencils of fabric'"

I was curious where this was heading; being a mathematics consultant, I knew what grade 2's were working on at this time of year (measurement). I didn't want to steal his thunder, so I just went with it.

"Really, Michael?" I turned into teacher mode: "Can you tell me some more about that?"

He went on to explain in great detail and with loads of enthusiasm about all the trouble he would run into if he wanted to measure fabric with random objects. He actually had a lot of fun naming all of the things that would be silly to use to measure fabric. He went on for a while and wrapped up the conversation telling me there was this "thing" called a "centimeter" that we could all use and understand. You would swear he discovered the metric system himself; he took such ownership of the concept.

Keep in mind, I can't think of a time he has ever been in a fabric store (I am not the crafty type) and I am almost certain that before this math lesson, he would never have used the word fabric (cloth or material, maybe?).

So he had no previous experience with the concept but he was still engaged? Yes.

​When I was at Dan Meyer's NCTM presentation (Beyond Relevance & Real World: Stronger Strategies for Student Engagement) last week, I couldn't help but think of this story from my son. I can imagine the kind of "teacher moves" my son's teacher used. She is a natural story teller, her enthusiasm is contagious and she loves to laugh. I can imagine her telling a story to the class, strategically leaving out important parts, having them experience her fabric store dilemma for themselves and brainstorming ideas with the class on how they can fix this problem!

Even if he didn't really discover the metric system, he certainly thought he did. And his teacher created those conditions. And I think that's pretty cool.

On the way home after a recent trip to the Seaport Market on the Halifax Harbour waterfront, I drove through the Port of Halifax Ocean Terminals. At Halterm, there is a shelter that trucks drive through to access the South End Container Terminal. This shelter contains an array of cameras that capture images of the truck and its cargo in order to identify it and quickly pass along information to the weigh station. The large door on this shelter has a sign on top with clearance height and width maximums.

This building allows trucks to speed up the movement of containers and make transport times more efficient. Given the size of the post-Panamax container ships that this port serves and the vast number of containers being moved, even a slight increase in speed can make a big increase in efficiency. There are lots of opportunities for students to pose interesting mathematical questions based on the photo above. Below are some of the questions that I thought about.

Image from Google Maps - http://bit.ly/1NzyAaW

1. Estimate (similar to http://www.estimation180.com/day-163.html) - I like that the clearance warning sign at the top of the door has both a maximum height and width listed. I removed the numbers for the maximums from the photo (attached below) and projected it for a group of students to look at (this was a combined class of Math at Work 11 and Math Essentials 11). I asked them to estimate how big this door is. The picture above showing the truck entering the door helped students refine their estimate. We also found online that a typical container truck like the one above is around 13.5 ft tall and 8.5 ft wide. They estimated in the range of 15-20 ft for the maximum height. Next I revealed that the height was 16 ft and asked them to then estimate the width given this new information. We measured the projected picture with a ruler and then used a ratio to calculate the width. We had a discussion regarding how close to the actual measurement the clearance might be.

2. Calculate - Lets assume that the clearance sign includes an extra 5% for safety. Given the clearances of 16 x 13 ft what would the actual dimensions and area of the door be? We could also calculate the volume or surface area of this shelter. It is very nearly a cube as can be seen from the image from Google Maps. There is one small door on the side and two large garage doors.

3. Convert - The clearances on the sign above the door are given in Imperial units, indicative of Canada's somewhat stalled process of metrication. What would these clearances be if they were converted to metric units of length. Which unit would be the best? The decimetre (dm) is the closest unit to feet but it is so rarely used Would it be a safety issue to use this measurement unit? If we use metres, how accurate should be make it? To the nearest metre? tenth of a metre? hundreth of a metre? How easy to read are the decimal points?

​4. Create - Given a door with dimensions of 16 ft by 13 ft, what is its area? How many other doors could you design that have this same area (if we constrain the dimensions to be whole feet)? Which of those doors could an average car drive through (for example a Chevy Malibu)?

5. Connect - Find a large garage door in your community that doesn't have a clearance indicated (such as at a fire station or service station). What would a clearance sign for this door say? What is the largest door that you can think of in Nova Scotia? Perhaps a hangar door at an airport or perhaps the door on the Ultra Hall of the Irving Shipyard? How big do you think the largest door is?

A nice extension is to explore the dangers of ignoring clearance warnings. There is an article from the Wall Street Journal called The Joys of Watching a Bridge Shave the Tops Off Trucks. The article reports on a bridge with low clearance in Durham, NC. Trucks crash into this bridge so regularly at an employee at a nearby office, Jürgen Henn who runs the website 11foot8.com, with a view of the bridge began to wonder: How often did this occur? “For weeks, you wouldn’t think about it,” Mr. Henn says. “Then there would be another one—and, oh, there’s another one.” An interesting statistics project would be to graph the dates of the crash to see how often they occur or if they are correlated to bad weather or season of the year or some other factor.

Nova Scotia Mathematics Curriculum Outcomes Mathematics 8 - M03Studentswill be expected todetermine the surface area of right rectangular prisms, right triangular prisms, and right cylinders to solve problems.Mathematics 9 - G01Students will be expected to determine the surface area of composite 3-D objects to solve problems.Mathematics at Work 10 - M03 Students will be expected to solve and verify problems that involve SI and imperial linear measurements, including decimal and fractional measurements. Mathematics 10 - M01Students will be expected to solve problems that involve linear measurement, using SI and imperial units of measure, estimation strategies, and measurement strategies. Mathematics Essentials 11 - D8 Estimate the volume and surface area using estimation strategies.Mathematics at Work 11 - M01Students will be expected to solve problems that involve SI and imperial units in surface area measurements and verify the solutions.

I did this activity recently with a group of Junior High math teachers. I started by showing them a picture of the square shape at right (from the Fraction Talks website) with one of the sections shaded. I asked them, "What fraction of this square is shaded?" Being teachers, they were quickly able determine that the shaded area was 3/16 of the total shape. Their answers however, were not nearly as interesting as how they got their answers. I asked a number of participants to explain their methods and reasoning.

There are a lot of different ways to get to the answer and it is great for participants to see the variety of ways that their peers tackled this problem. One participant described how she broke the shape up into sixteen small equal triangles and and found that the shaded area was 3 of these 16 small triangles. Another saw that the diagonal from the top left to the bottom right cut the triangle in half. The large triangle on the bottom is 1/4 and the small triangle in the top left is 1/16. This made the shaded region 1/2 - 1/4 - 1/16 = 3/16.

We also discussed the likelihood that a student might say that 1/7 of the shape is shaded. A student might see this a 1 of the 7 pieces of this shape is shaded. This would be thinking about the number of pieces instead of the area of the pieces and is something they may have done with Tangrams or pattern blocks in earlier grades.

There are lots of questions you can use to extend this discussion. (It might help to label each of the smaller shapes in the square with a letter or hand out a picture of the square for students to work with.)

How could you shade twice this area (i.e. 3/8 of this square)?

How many different ways can you find to shade exactly one-quarter of the shape?

​If the entire square has an area of 12 square units, what is the area of the shaded section?

If the shaded section has an area of 6 square units, what is the area of the entire square?

Once everyone was familiar with the idea of creating and describing fractions by shading the square, I handed out a piece of paper to each person (or pairs if you have a big group) that could be folded an hung up on our clothesline number line. The paper tent had the square and a place to write a fraction on one side and some instructions on the other side. The instructions were:

Once everyone had a chance to shade their square and determine their fraction, I had them hang their paper in the appropriate spot on the clothesline. We determined a few benchmarks: The left end of the clothesline was 0 and the right end was 1. A spot in the center of the clothesline was 1/2. The next time I do this activity, I think it might be helpful to create some benchmark cards to hang on the number line. When participants had different ways of shading the same fraction, we used clothes pins to connect them together on the clothesline to show that they were equivalent.

After everyone had placed a paper tent on the clothesline, I asked if we had found all of the fractions that could be shaded. I handed out a few additional blank paper tents and challenged them to find a way to shade a fraction that wasn't already on our clothesline. As a bit of enrichment, you could ask participants to find a way to calculate or reason how many total fractions are possible with this shape in order to prove if we had found them all or not yet. This was a really fun activity and there was lots of engagement.

The Riddle of the Tiled Hearth is one of many mathematical puzzles from Henry Ernest Dudeney's 1907 book titled The Canterbury Puzzles And Other Curious Problems. The first group of puzzles in this book are based on the characters from Geoffrey Chaucer's Canterbury Tales. Puzzles from this book could be used as part of a cross curricular unit on history, literature and mathematics. There are a number of very interesting puzzles and games including the first pentomino puzzle called The Broken Chessboard and a clever variation of the game Nim called The Thirty One Game. Since this book was first published in 1907, the copyright has expired and it is freely available on Project Gutenberg.

It seems that it was Friar Andrew who first managed to "rede the riddle of the Tiled Hearth." Yet it was a simple enough little puzzle. The square hearth, where they burnt their Yule logs and round which they had such merry carousings, was floored with sixteen large ornamental tiles. When these became cracked and burnt with the heat of the great fire, it was decided to put down new tiles, which had to be selected from four different patterns (the Cross, the Fleur-de-lys, the Lion, and the Star); but plain tiles were also available. The Abbot proposed that they should be laid as shown in our sketch, without any plain tiles at all; but Brother Richard broke in,--"I trow, my Lord Abbot, that a riddle is required of me this day. Listen, then, to that which I shall put forth. Let these sixteen tiles be so placed that no tile shall be in line with another of the same design"—(he meant, of course, not in line horizontally, vertically, or diagonally)—"and in such manner that as few plain tiles as possible be required." When the monks handed in their plans it was found that only Friar Andrew had hit upon the correct answer, even Friar Richard himself being wrong. All had used too many plain tiles.-The Canterbury Puzzles by Henry Ernest Dudeney 1907

To use this activity with students, I would start by introducing using the tiled hearth story as written above. Then I would introduce some manipulatives that would let them physically explore and work with the puzzle. I would give each group of students a large 4x4 grid on a sheet of paper and some multi-link cubes of 4 different colours.

The goal for students is to place as many of the cubes as possible onto the paper such that there is one cube per square and there can be no cubes of the same colour in any row, column or diagonal. I might start by showing them a few different incorrect solutions and ask them what is wrong with that solution. This would help students understand the constraints. Then I would let them explore for a while.

One aspect of this puzzle that I like is that students can play around with it and have some intermediate success. They might just place a few cubes on the grid. With time, they can refine their solutions to get better and better. Below shows how a student might explore to place more and more cubes.

The Solution from Canterbury Puzzles shows that the best solution leaves 3 blank spaces. Dudeney states, "The correct answer is shown in the illustration on page 196. No tile is in line (either horizontally, vertically, or diagonally) with another tile of the same design, and only three plain tiles are used. If after placing the four lions you fall into the error of placing four other tiles of another pattern, instead of only three, you will be left with four places that must be occupied by plain tiles. The secret consists in placing four of one kind and only three of each of the others." Below are both my solution using cubes and Dudeney's equivalent solution.